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Feynman's "no-node" theorem states that the conventional many-bodyground-state wavefunctions of bosons in the coordinate representation ispositive-definite. This implies that time-reversal symmetry cannot bespontaneously broken. In this article, we review our progress in studying aclass of new states of unconventional Bose-Einstein condensations beyond thisparadigm. These states can either be the long-lived metal-stable states ofultra-cold bosons in high orbital bands in optical lattices as a result of the"orbital-Hund's rule" interaction, or the ground states of spinful bosons withspin-orbit coupling linearly dependent on momentum. In both cases, Feynman'sargument does not apply. The resultant many-body wavefunctions arecomplex-valued and thus break time-reversal symmetry spontaneously. Exoticphenomena in these states include the Bose-Einstein condensation at non-zeromomentum, the ordering of orbital angular momentum moments, the half-quantumvortex, and the spin texture of skyrmions.

Unconventional Bose-Einstein Condensations Beyond the "No-node" Theorem